# Are there fields (of any kind) inside a black hole?

It is said that nothing escapes from black holes, not even light. All particles are now thought to be excitation of different fields (electric field, electromagnetic field, photon field, etc).

Does it follow that there are no fields (of whatever kind) inside the event horizon of a black hole? IFFY (if and only if) there are fields, are particles created inside the black hole?

• The abbreviation for "if and only if" is "iff", not "IFFY". Commented Feb 13, 2021 at 1:06
• Why might what escaped from black holes say anything about what happened inside? Does no light escaping mean no light inside, for instance? Commented Feb 13, 2021 at 1:36
• @safesphere I've heard some say that there's no space time inside the event horizon. What does this mean? and, on the basis of this assumption, what are its implications? Commented Feb 17, 2021 at 14:35

Sure, the reisner-Nordström solution is given by:

$$ds^{2} = -f(r)dt^{2} + \frac{1}{f(r)}dr^{2} + r^{2}d\theta^{2} + r^{2}\sin^{2}\phi^{2}$$

where

$$f = 1 -\frac{2M}{r} + \frac{q^{2}}{r^{2}}$$

This is a solution to the Einstein-Maxwell equations with 4-vector potential:

$$A_{a}ds^{a} = \frac{q}{r}dt$$

So, this solution has an electric field that is nonzero everywhere, including inside the horizon.

• Thanks! So after everything, not all is lost with black holes. That's an encouraging answer. Commented Feb 12, 2021 at 15:59
• On second thought, a varying electric field creates a magnetic field which is tantamount to an electromagnetic field. Is it then possible to have an electromagnetic field inside the black hole? Commented Feb 12, 2021 at 16:06
• @RoyClosa: the spinning charged black hole solution has a vector potential with an angular component that is nonzero inside the horizon, but it also has many more terms and I can't write it down from memory. Commented Feb 12, 2021 at 16:07
• And actually, now that I think of it, since the time coordinate is spacelike inside the horizon here, this given vector potential would describe a magnetic field inside the horizon. Commented Feb 12, 2021 at 16:28

We don't know for certain what is inside the event horizon of a black hole, but we expect there to be fields inside the event horizon that are just extensions of the fields outside of the event horizon. The only restriction is that events that happen inside the event horizon cannot be causally linked to effects outside the event horizon.

However, a complete description of the event horizon at the very smallest scales requires a theory of quantum gravity, which we do not yet have.

Besides the electric field, as Jerry Schirmer mentions in his answer, that is non-zero everywhere one can have a scalar field that is exists inside the horizon. See for example the (2+1) dimensional solution reported here Conformally dressed black hole in 2+1 dimensions.

The metric function is found to be:

$$f(r) = \cfrac{(r+B)^2(r-2B)}{rl^2}$$

and the scalar field:

$$\Psi = \sqrt{\cfrac{8B}{\kappa(r+B)}}$$

where $$B>0$$. The black hole horizon is located at $$r_h=2B$$ and the scalar field remains finite for all $$r>0$$.